HGeometry 5-4 Lesson ASA and AAS (PDF)

## **5-4 Proving Triangles Congruent: ASA & AAS** **Learning Targets** * I can use the ASA Congruence criterion for triangles to solve problems and prove relationships in geometric figures. * I can use the AAS Congruence criterion for triangles to solve problems and prove relationships in geometri...

## **5-4 Proving Triangles Congruent: ASA & AAS** **Learning Targets** * I can use the ASA Congruence criterion for triangles to solve problems and prove relationships in geometric figures. * I can use the AAS Congruence criterion for triangles to solve problems and prove relationships in geometric figures. **Vocabulary** * **Included Side:** An included side is the side of a triangle between two angles. In ∆ABC, AC is the included side between ∠A and ∠C. **EE1: Use ASA to Prove Triangles Congruent** **Given:** QS bisects ∠PQR. ∠PSQ ≅∠RSQ **Prove:** ∆POS ≅∆ROS 1. QS bisects ∠PQR. Given 2. ∠PSQ ≅∠RSQ Given 3. ∠PQS ≅∠ROS Definition of angle bisector 4. OS ≅ OS Reflexive 5. ∆POS ≅∆ROS ASA **Proving Triangles Congruent: ASA** In Lesson 5-2 you learned that two triangles are congruent if two pairs of sides are congruent and the included angles are congruent (SAS). It is also true that two triangles are also congruent if two pairs of angles are congruent and the included sides are congruent (ASA). **Postulate 5.3: Angle-Side-Angle (ASA) Congruence** If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. If ∠A ≅ ∠D, AB ≅ DE, and ∠B ≅ ∠E, then ∆ABC ≅∆DEF. **Proving Triangles Congruent: AAS** **Theorem 5.5: Angle-Angle-Side (AAS) Congruence** If two angles and the nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. If ∠A ≅ ∠D, ∠B ≅ ∠E, and BC ≅ EF, then ∆ABC ≅∆DEF. **EE3: Use AAS to Prove Triangles Congruent** **Given:** R is the midpoint of QS. ∠QPR ≅∠STR. ∠Q and ∠S are right angles. **Prove:** ∆PQR ≅∆TSR 1. R is the midpoint of QS Given 2. ∠QPR ≅∠STR Given 3. ∠Q and ∠S are right angles. Given 4. QR ≅SR Definition of midpoint 5. ∠Q ≅∠S Right angles are congruent 6. ∆PQR ≅∆TSR AAS **Given:** RQ ≅ ST. RQ || ST **Prove:** ∆RUQ ≅ ∆TUS 1. RQ ≅ ST Given 2. RQ || ST Given 3. ∠RUQ ≅∠TUS AIAT 4. ∠RQU ≅∠TSU VAT 5. ∆RUQ ≅∆TUS AAS **Homework** **Edulastic** HG ASA and AAS Proof Check